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A116664 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k parts that appear exactly once (n>=0, k>=0).

%I A116664 #9 May 13 2015 05:10:18
%S A116664 1,0,1,1,0,1,1,1,0,1,1,2,0,2,1,1,1,4,0,2,2,2,3,4,0,1,3,4,3,0,3,7,1,1,
%T A116664 5,4,6,0,4,10,2,2,6,7,9,0,7,12,5,3,7,13,11,0,1,8,18,7,5,0,11,15,18,1,
%U A116664 1,10,25,11,8,0,13,23,24,2,2,15,32,16,13,0,16,33,32,5,3,18,43,24,19,0,23,40
%N A116664 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k parts that appear exactly once (n>=0, k>=0).
%C A116664 Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0)=A097242(n). Sum(k*T(n,k), k>=0)=A116665(n).
%H A116664 Alois P. Heinz, <a href="/A116664/b116664.txt">Table of n, a(n) for n = 0..600, flattened</a>
%F A116664 G.f.: product(1+tx^(2j-1)+x^(4j-2)/(1-x^(2j-1)), j=1..infinity).
%e A116664 T(10,2) = 3 because the only partitions of 10 into odd parts and having exactly 2 parts that appear only once are [9,1],[7,3] and [5,3,1,1].
%e A116664 Triangle starts:
%e A116664 1;
%e A116664 0, 1;
%e A116664 1, 0;
%e A116664 1, 1;
%e A116664 1, 0, 1;
%e A116664 1, 2, 0;
%e A116664 2, 1, 1;
%e A116664 1, 4, 0;
%p A116664 g:=product(1+t*x^(2*j-1)+x^(2*(2*j-1))/(1-x^(2*j-1)),j=1..30): gser:=simplify(series(g,x=0,30)): P[0]:=1: for n from 1 to 25 do P[n]:=coeff(gser,x^n) od: for n from 0 to 25 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form, with one extra 0 in some rows
%p A116664 # second Maple program:
%p A116664 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A116664        expand(add(b(n-i*j, i-2)*`if`(j=1, x, 1), j=0..n/i))))
%p A116664     end:
%p A116664 T:= n-> (p-> seq(coeff(p, x, k), k=0..floor(sqrt(n))))
%p A116664         (b(n, n-irem(n+1, 2))):
%p A116664 seq(T(n), n=0..25);  # _Alois P. Heinz_, Mar 16 2014
%t A116664 b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[Sum[b[n-i*j, i-2]*If[j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, k], {k, 0, Floor[Sqrt[n]]}]][b[n, n-Mod[n+1, 2]]]; Table[T[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)
%Y A116664 Cf. A000009, A097242, A116665.
%K A116664 nonn,tabf
%O A116664 0,12
%A A116664 _Emeric Deutsch_, Feb 22 2006